The numbers \(x\), \(y\) and \(z\) are located on the real line as shown in the figure:
We can state that:
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\( xy < xz \)
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\( xy > z \)
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\( y^2 < x^2 \)
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\( x^2 < z^2 \)
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\( xy < z \)
Try to solve it before checking the answer.
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\( y^2 < x^2 \)
The illustration shows that
Since they are inequalities we have that:
\( x < y, \quad x < 0 \Rightarrow x^2 > xy \hspace{4em} \boldsymbol{(1)} \)
\( x < y, \quad y < 0 \Rightarrow xy > y^2 \hspace{4em} \boldsymbol{(2)} \)
From (1) and (2), by transitivity, we have that \( x^2 > y^2 \). In other words:
From (1) and (2), by transitivity, we have that \( x^2 > y^2 \). In other words, \(\boldsymbol{ y^2 < x^2 } \)